Homogenizing coils for nmr apparatus

ABSTRACT

A nuclear magnetic resonance apparatus adapted for homogenizing a field includes electrically insulated electroconductors positioned on pairs of plates and arranged for generating an incremental magnetic field in the vicinity of the sample and in the direction of the main field. The incremental field is represented by spherical harmonic functions. The incremental fields are substantially orthogonal for primary degree np and order mp. Arcuate and radial segments of the electroconductors are positioned on opposite surfaces of a plate with connections through the plate. The arcuate segments are radially spaced from a center thereof in a manner for reducing to a relatively low level, ancillary interferring harmonics which are also generated by the electroconductors. Means are also provided for combining orders of spherical harmonic functions to effect additional correction of interferring ancillary harmonics and to homogenize the field of a spinning sample.

United States Patent [72] Inventor Marcel J. E. Golay 116 Ridge Road,Rumson, N .J 07760 [21 Appl. No. 733,522 [22] Filed May 31, 1968 [45]Patented Nov. 23, 1971 Continuation-impart of application Ser. No.649,539, June 28, 1967, now abandoned. This application May 31, 1968,Ser. No. 733,522

[54] HOMOGENIZING COILS FOR NMR APPARATUS 884,129 l2/l96l GreatBritaint.

OTHER REFERENCES l. Zupancic, Current Shims For High Resolution NuclearMagnetic Resonance On The Problem Of Correcting Magnetic Fieldinhomogeneities, Journal of Scientific Instruments. 39(12), December1962, pp. 62 l- 624.

Primary Examiner-Michael J. Lynch AIt0rney-Edward R. Hyde, Jr.

ABSTRACT: A nuclear magnetic resonance apparatus adapted forhomogenizing a field includes electrically insulated electroconductorspositioned on pairs of plates and arranged for generating an incrementalmagnetic field in the vicinity of the sample and in the direction of themain field. The incremental field is represented by spherical harmonicfunctions. The incremental fields are substantially orthogonal forprimary degree n, and order m,,. Arcuate and radial segments of theelectroconductors are positioned on opposite surfaces of a plate withconnections through the plate. The arcuate segments are radially spacedfrom a center thereof in a manner for reducing to a relatively lowlevel, ancillary interferring harmonics which are also generated by theelectroconductors. Means are also provided for combining orders ofspherical harmonic functions to effect additional correction ofinterferring ancillary harmonics and to homogenize the field ofaspinning sample.

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HOMOGENIZING'COILS FOR NMR APPARATUS This application is acontinuation-in-part of US. Pat. application, Ser. No. 649,539, filedJune 28, i967, now abandoned.

This invention relates to nuclear magnetic resonance (NMR) apparatusadapted for analyzing substances by NMR techniques and more particularlyto means for improving the uniformity of a magnetic field established bysuch apparatus.

NMR techniques have been employed in apparatus which function toidentify a substance by an atomic analysis of the substance. in general,a sample under investigation is positioned a relatively intense (1,000to 23,000 Gauss) magnetic field of amplitude, H,. The Larmour frequencyof the atomic nucleus of the substance is determined in one form of NMRinstrument by superimposing a less intense alternating magnetic field Hof variable frequency on the steady field and noting the frequency ofresonance. The gyromagnetic ratio (l/u) of a sample element, where 1represents the nuclear angular momentum and u represents the magneticmoment of the nucleus, is thus determinable from a knowledge of thequotient of the intensity of the field, H,, over the Larmour frequency(f An identification of the particular element and isotope can thereforebe realized.

Microanalytical apparatus of this type require that a highly homogeneous(i.e., uniform) field, H,, exist in order that an accurateidentification of the nuclei can be made. Nonuniformities in the fieldof an order as small as in the vicinity of the sample can interfere withaccurate identification. Prior NMR arrangements have employedfield-correcting electroconductors, also termed field-homogenizing orshim coils, which generate corrective fields for improving fielduniformity in the vicinity of the sample.

In an arrangement for homogenizing the field, one or more pairs ofrelatively flat electroconductors are positioned in a gap between twopole faces of a magnet which establishes the field H,. The field ishomogenized by manually varying the amplitude of currents flowing in theelectroconductors until an acceptable field uniformity is established.These current adjustments are repetitive and become tedious andtime-consuming because of interactions occurring between the fieldsestablished by the different electroconductors. In order to reduce thenumber of such current adjustments, each electroconductor has beenarranged in a manner for generating an incremental magnetic field in thevicinity of the sample and in the direction of H,. This incrementalmagnetic field can be represented by a spherical harmonic function. Theorthogonal (i.e., independent) relationships existing between thespherical harmonic functions permit field-correcting current amplitudeadjustments to be made in one electroconductor which theoretically willnot alter the effectiveness of the field established by current flowingin another electroconductor. The number of current adjustments requiredto attain a desired degree of homogenization is thereby reduced.

In an ideal situation, a spherically shaped sample is placed in thecenter of coordinates in the gap center and the sample is uniformlyradiated by the AC field H Then, it is sufficient, in principle, toadjust the current in each coil only once. ln practice, however, thesample may be massive, chunky, instead of spherical and the radiatingfield may depart from uniformity, in which case a typical arrangementwill provide relatively good convergence towards an optimum setting ofthe currentamplitude-controlling means.

In order to provide a high-intensity uniform field and to avoid leakageflux phenomenon, it is desirable that the gap distance between the polefaces of the magnet producing the field H be maintained at as small avalue as is physically possible. The limiting value of this distance isdetermined by the dimensions of the apparatus required to support thesample and various coils in the gap. In general, the degree ofhomogenization provided is dependent upon the number of homogenizingcoils employed. The use of many shim coils, one on each insulated sheet,in order to provide suitable homogenization, increases appreciably therequired gap distance. Since the gap distance should be relativelysmall, a

design compromise is drawn between the gap distance and the extent offield homogenization provided.

It is one object of this invention to provide a fieldhomogenizing meanswhich occupy a reduced magnet gap distance while providing an increaseddegree of field homogenization.

Another object of this invention is to provide an improved arrangementof substantially noninteracting current-carrying homogenizing coils ineach of which the currents can be adjusted to optimize some observablequantity such as the line width of some NMR line in aspectrogram,substantially independently of the adjustment of current in the othercoils.

Another object of the present invention is to provide an improvedarrangement for homogenizing the field of a nuclear magnetic resonanceapparatus.

A further object of the invention is the provision of a plurality ofplanar homogenizing electroconductors adapted for generating a pluralityof spherical harmonic functions and which are arranged in asubstantially regular configuration.

Still another object of the invention is to provide a fieldhomogenizingarrangement having a physical configuration which facilitates theaddition of electroconductors for increasing the degree ofhomogenization provided.

In prior apparatus, the homogenization of the field is effectivelyincreased by spinning the sample under investigation about an axis.However, a spinning sample is also accompaniecl by modulation affects ofa spherical harmonic order which at times are not correctable by shimcoils adapted for use with a stationary sample.

It is another object of this invention to provide improved means foreffecting homogenization of the magnetic field in the presence of eithera stationary or a spinning sample.

Another object of the invention is to adapt a homogenizing means, whichis provided for homogenizing the field about a stationary sample, in amanner for homogenizing the field about a spinning sample.

In accordance with the more general features of the present invention,an NMR apparatus adapted for homogenizing a field includes electricallyinsulated electroconductors positioned on pairs of plates and arrangedfor generating an incremental magnetic field in the vicinity of thesample and in the direction of the main field. The incremental field canbe represented by spherical harmonic field functions referred to asystem of coordinates in which the polar axis of the system is normal tothe magnet pole pieces. These incremental fields are substantiallyorthogonal for their primary degree ri and order m,, i.e., for thedegree and order of the potential function they are specificallydesigned to generate. The electroconductors are supported by a singlepair of plates, the individual plates of the pair being positioned atopposite pole faces of the device. The electroconductors include arcuateand radial segments respectively positioned on opposite surfaces of theplate with connections through the plate, the arcuate segmentsbeing'radially spaced from a center thereof in a manner for reducing toa predetermined relatively low level, some of the ancillary interferingharmonics which are also generated by the electroconductors. Circuitmeans are additionally provided for effecting cancellation of theremaining interfering ancillary harmonics. In a particular arrangement,a plurality of such electroconductors are formed as relatively thin fiatconductors by printed circuit techniques. Thus, the arcuate and radialelectroconductor configuration provides on pairs of plates,electroconductors adapted for generating a plurality of primaryspherical harmonics. In this manner, a relatively large degree of fieldhomogenization is advantageously provided by a configuration occupying arelatively small portion of the gap distance.

In accordance with another feature of the present invention, means areprovided for combining orders of spherical harmonic functions which areadapted for homogenizing the field of a stationary sample, in a mannerfor homogenizing the field of a spinning sample.

These and other objects and features of the invention will becomeapparent with reference to the following specifications and drawingswherein:

FIG. I is a diagram illustrating a nuclear magnetic resonanceinstrument;

FIG. 2 is an enlarged view of the gap in a magnet of the instrument ofFIG. 1 illustrating an assembly of plates upon which are supportedhomogenizing electroconductors fabricated in accordance with the presentinvention;

FIG. 3 is a diagram illustrating one plate of a first pair of plateshaving a plurality of homogenizing electroconductors constructed inaccordance with the present invention;

FIG. 4 is a diagram illustrating one plate ofa second pair of plateshaving a plurality of homogenizing electroconductors constructed inaccordance with the present invention;

FIGS. 5, 7, 9, 11,13, 15,17, 19, 2I, 23 and 25 are diagrams illustratingthe locus of current paths on a sphere of electrically insulativematerial and which generate magnetic fields at the center of the spherewhich are represented by spherical harmonic functions;

FIGS. 6, 8, l0, l2, l4, l6, I8, 20, 22, 24 and 26 are diagramsillustrating generally the projection of current paths which have themorphology of the current paths of FIGS. 5, 7, 9, ll, 13, 15, 17, I9,21, 23 and 25 respectively projected on pole faces of the magnet of theinstrument of FIG. 1;

FIG. 2-7 is a diagram utilized for determining the magnitude ofancillary harmonic functions at a point in the magnet gap;

FIG. 28 is a diagram ofa portion of an electroconductor for establishinga spherical harmonic corrective field of order m=l and illustrating asemicircular configuration of arcuate conductive segments;

FIG. 29 is a diagram illustrating the resolution of the electroconductorof FIG. 28 into separate semicircular components;

FIG. 30 is a diagram illustrating the semicircular arrangement of aportion of an electroconductor for generating a spherical harmonicfunction of order m=l and the resolutions thereof into separatesemicircular components;

FIG. 31 is another diagram utilized for determining the magnitude ofancillary interfering harmonic functions;

FIG. 32 is a diagram ofa portion of an electroconductor for generating aspherical harmonic function of order and illustrates the quartercircular configuration of arcuate electroconductor segments and theirresolution into separate building blocks;"

FIG. 33 is a diagram ofa portion of an electroconductor for generating aspherical harmonic of degree n=4 and order "i=3;

FIG. 34 is a diagram illustrating a circuit for cancelling ancillaryharmonics not cancelled by spacing of arcuate coil segments;

FIG. 35 is a diagram illustrating a portion of a spectrogram generatedby a NMR apparatus when a sample under investigation is subjected tospinning;

FIG. 36 is a diagram of a circuit arrangement adapted for combiningcurrents, which generate harmonic functions, in a manner for providingdesired field-corrective functions for a spinning sample;

FIG. 37 is a diagram of a circuit arrangement adapted for combiningcurrents, which generate harmonic functions, in a manner for providingadditional desired field-correcting functions for a spinning sample, and

FIG. 38, 39, 40 and 41 is a perspective view ofa sphere having windingspositioned thereon and helpful in visualizing current paths.

To a large extent, the field-corrective considerations applicable to astationary sample are similarly applicable to a spinning sample.Accordingly, the following discussion will initially describe thegeneration of homogenizing coils with respect to a stationary sampleand, subsequently, the generation of homogenizing coils with respect toa spinning sample.

Although the use of mathematical spherical harmonic func tions withrespect to the homogenizing fields generated in NMR apparatus is known,it is believed that features of the present invention may be betterunderstood from the following simplified considerations of thesefunctions. The component H of the magnetic field intensity in thedirection of the polar axis z, established at a point in a gap betweenpole faces of the magnet, has the property that its Laplacian vanishes.Stated mathematically,

This same property is applicable to a distorting inhomogeneous fieldwhich, if added to a uniform field having the average intensity anddirection of the actual field at the airgap center, produces the actual,inhomogeneous field in the region of interest near the airgap center. Itis also applicable to individual incremental corrective fields which canbe generated by homogenizing coils traversed by electrical currents. Itis known, also, that in the immediate neighborhood of any point and inparticular of an origin of coordinates any distorting field can beexpressed as the sum of incremental fields each of which isrepresentable by a spherical harmonic referred to a polar coordinatesystem centered in said origin of coordinates in which the z or polaraxis passes through the airgap center and is normal to the pole faces,said spherical harmonies possessing no singularity in said origin. Suchspherical harmonics satisfy equation (I). In practice, the effect of theperpendicular distorting fields is negligibly small. Therefore, if amultiplicity of electroconductors is provided, each of which whentraversed by electrical current produces an incremental field with acomponent parallel to the main field which is expressed substantially bya spherical harmonic, then it is possible to decrease the inhomogenitiesin the initial field produced by the magnet by means of substantiallyorthogonal (independent) current controls. Furthermore, the greater thenumber of individual electroconductors of the type just described thatare provided, the greater will be the corresponding number of sphericalharmonic functions generated by these electroconductors. Consequently, acorrective field can be more closely approximated by the superpositionof these harmonic functions and the field homogenization accomplishedwill be the greater. Although, for ease of description, anelectroconductor is referred to in the specification as generating aspherical harmonic, it will be understood that the current-carryingelectroconductors establish magnetic fields in the z direction as wellas more generally, magnetic potentials, which are expressible asspherical harmonics.

Spherical harmonic functions can be derived in a wellknown manner fromthe Legendre functions. The spherical harmonic functions describingvarious incremental fields parallel to the main field are of the forms:

first degree harmonic function: 2, .r and y (2) second degree harmonicfunction: 2z x y .rz, yz, x y,

-y third degree harmonic function: 2z'-'3z(x +y x[4z (x y y[4z'-(.x +y(X -31 z, xyz, .r3xy and 3x Y-y The first degree fields can be obtained(neglecting an inconsequential numerical factor) as the differentiationwith respect to z of the zonal and two tesseral harmonic functions:

2z (.r +y zx, and 2y, (3) which will be designated hereinafter by thecodes (2,0), (2,1 and (2,1 respectively.

Similarly, the second degree fields can be derived from the zonal andtesseral harmonic function of third degree and will have the codes:(3,0); (3,l); (3,l)'; (3,2); (3,2); etc., and more generally will havethe codes (me) for zonal harmonics and (n,m) or (n,m) with the n m o fortesseral harmonics, sectorial harmonics with the code (n,n) not beingconsidered here because they correspond to fields perpendicular to themain field and of little importance as explained earlier. In general, ndesignates the degree of the harmonic potential from which a field ofdegree (n-l) can be obtained by differentiation with respect to z and mdesignates the order ofthe harmonic potential which is also the order ofthe spherical harmonic of the field derived from it. Since sectorial andtesseral harmonic functions, (i.e., m occur in pairs, then hereinafterthe second of the pair will be distinguished from the first of the pairby a prime symbol; e.g., (2,!) and (2,1); and generally, (n,m) and(n,m)'.

A rigorous configuration for generating the spherical harmonic functionscomprises a plurality of continuously distributed surface currentsflowing on a sphere which is centered about the sample point in themagnet gap. However, the placement of a sphere in the magnet gap wouldphysically interfere with the placement of the sample under analysis inthe gap and continuously distributed currents on a surface aretechnically unfeasible. A practical arrangement for field correction hastaken the form of a group of electrically insulative thin flat platesmounted near each of the pole faces in the gap, each of which supportsthin flat electroconductors generating substantially, in the region ofinterest, a field represented by a desired spherical harmonic functionof degree (n) and order (m). Because of this planar configuration andthe finite dimensions of the current-carrying electroconductors, otherharmonics in addition to the desired spherical harmonic of degree (n)and order (m) will be generated. Hereinafter, the spherical harmonicswhich it is desired to generate are referred to as primary sphericalharmonics while the undesired additionally generated harmonics arereferred to as ancillary harmonics. When the electroconductors aresymmetrically arranged as indicated hereinafter, the undesired ancillaryharmonics are of an order which is an odd multiple of m: 3m, 5m etc., orhave a degree which differs from n by an even integer, or both. It isanother object of the present invention to provide homogenizing coils inwhich several of the ancillary harmonics are reduced to a negligiblevalue.

The NMR instrument illustrated in FIG. I includes a magnet for producinga main field of desired intensity, H,. This magnet, of the electromagnettype, comprises a ferromagnetic core 9 having a gap between pole faces10 and 11 thereof and a winding 12. A current source 14 for causing acurrent to flow in the winding 12 is provided for establishing themagnetic field of intensity H Alternatively, the magnet may comprise apermanent magnet rather than an electromagnet. A sample under analysisis positioned in the gap and supported by conventional means, indicatedgenerally by the rectangle 20. This support means houses in addition, afield coil (not shown) for establishing an alternating field ofintensityH in a plane perpendicular to the field H of the magnet as well as aninductive pickup coil (not shown) having an axis orientatedperpendicularly both with respect to the field H, and the field H Thesupport means 20 is adapted for spinning the sample under investigationor for supporting it in a stationary mode. An RF signal generator 22 iscoupled to the field coil for generating a field H This field coil isexcited at an RF frequency, or over a range of RF frequencies, while asignal receiver and indicating means 24 is coupled to the pickup coil inthe gap for detection and indication of nuclear magnetic resonance.

In operation, the Larmour frequericyf of the sample under investigationis determined by establishing a steady field H and varying theexcitation frequency f, of the field H until a resonance is determined,as evidenced by a greatly increased signal output from the pickup coil.Alternatively, the RF frequencyf is held constant and the magnitude ofthe field H is cyclically varied, such as at a 60-cycle rate. Thegyromagnetic ratio (I/u) is related to the ratio of the field H over theLarmour frequency 11,, and thus may be calculated. This general NMRapparatus and technique is well known and further elaboration is notdeemed necessary.

In accordance with features of this invention, a plurality of relativelythin field-homogenizing electroconductors are supported by first andsecond plates at opposite pole faces 10 and 11 respectively and areadapted for generating correction fields represented by an equalplurality of primary spherical harmonic functions. The generalarrangement of the plates in the magnet gap is illustrated in FIGS. 1and 2 while a particular configuration of the electroconductors isillustrated in FIGS. 3 and 4. As indicated in greater detailhereinafter, associated electroconductors are supported by the first andsecond plates for generating a primary spherical harmonic function. A DCcurrent is caused to fiow in electroconductors of each of these platesfrom an adjustable current source, indicated generally as 30. In aparticular arrangement of the present invention, as illustrated in FIGS.3 and 4, two pairs of plates provide 17 spherical hannonic functions, l6of which are corrective, which is equivalent, generally, to the numberof corrective functions previously provided by l6 pair of plates. Oneelectroconductor of the group of 17 is provided for varying the mainfield by a minute amount, when required by the particular NMRarrangement employed.

FIG. 2 is an enlarged view of the magnet gap of FIG. 1 illustrating ingreater detail an arrangement of the two pairs of plates which arefabricated by printed circuit techniques. A first plate of a pairpositioned near pole face 10 comprises a thickness of insulatingmaterial 32, such as Mylar 0.005 inches in thickness, supporting onopposite surfaces thereof, two thin metal foils 34 and 36, such ascopper 0.001 inches in thickness. An arrangement of this type maycomprise a double-clad printed circuit board. The other plate of thepair positioned near pole face 11 comprises a double-clad circuit boardhaving a thickness of insulating material 38 and metal foils 40 and 42.Electroconductors are formed from the metal foils of the plate in amanner indicated in greater detail hereinafter. This pair of plates iselectrically insulated from the metallic pole faces 10 and II by Mylarinsulating material 44 and 46 respectively, and are secured by anysuitable means such as a nonmagnetic, adhesive material. An epoxy resinglue is a typical nonmagnetic adhesive material. Associatedelectroconductors on the pair of plates which operate to generate aparticular spherical harmonic function are coupled in series by leadsindicated generally by the wires 48. Currentfiows in theelectroconductors, for each function, from thecurrent-amplitude-adjusting means 30 which includes the sources ofpotential 49 and 50 and potentiometer S]. A second pair of platescomprises the double-clad printed circuit boards formed by the insulator54 and metal foils 56 and 58 and by the insulator 60 and metal foils 62and 64. The electroconductors formed on this pair of plates aresimilarly coupled in series by leads 66 and current flows therein fromthe potential sources 49 and 50 via a potentiometer 52 of thecurrent-amplitude-adjusting means 30. The adjacent plates of both pairsare insulated from each other by insulators 61 and 63.

FIG. 3 illustrates the electroconductor arrangement of one plate of apair of field-homogenizing plates while FIG. 4 illustrates theelectroconductor arrangement of one plate of a second pair offield-homogenizing plates. For purposes of clarity in the drawings,these electroconductors are shown enlarged on the order of three to fivetimes their actual dimensions. Similar plates of the pairs are providedfor mounting adjacent the opposite pole face of the magnet. The plate ofFIG. 3 is represented in FIG, 2 by the double-clad board comprising theinsulator 32 and foils 34 and 36 while the plate of FIG. 4 isrepresented by the insulator 54 and foils 56 and 58. It can be seen fromFIGS. 3 and 4 that these electroconductors include generally arcuatesegments, shown in solid line, formed from the metal on one side of theplate and radial segments, shown in dashed line, formed from the metalon an opposite side of a same plate. Conductive connections are madethrough the insulative material between the arcuate and radial segmentsas indicated in FIGS. 3 and 4. The arcuate electroconductors associatedwith a particular primary spherical harmonic function are positionedspatially in a manner for generating substantially orthogonal correctionfor the primary degree and order (n,,, m in the vicinity of the samplewhile reducing the generation of ancillaryspherical harmonics at thesame point in the gap.

The electroconductor configurations of FIGS. 3 and 4 are generated inthe following manner. When a conductor, traversed by a current, ispositioned on the surface of an insulative sphere and along the locus ofa particular spherical harmonic function where the function vanishes,with the current flowing in the direction faced by an observer havingthe positive values of the function on his left and negative on hisright, the sphere being centered about the sample point in the magnetairgap and the electroconductor having a unity current flowing therein,then a magnetic field is generated which, at the center of the sphere,approximates to a relatively high degree, said spherical harmonicfunction. By projecting the several geometrical configurations of thecurrent flow loci as viewed from the center of the insulative sphere tothe flat magnet pole faces, groups of concentric circles as well asradial spokes are obtained on a common plane. These projectedconfigurations of the current flow loci generate fields at the center ofthe current flow loci generate fields at the center of the sphere whichare orthogonally related for what will be termed the primary degree upand primary order mp: (21,, m

Although an electroconductor arrangement for establishing fieldsrepresented by spherical harmonics of primary degree and order (n,,, m,)is theoretically achieved in this manner, such an arrangement ofelectroconductors of finite dimensions in a common plane formed togenerate a particular desired spherical harmonic of primary degree andorder (n,,, m,,) also generates ancillary, i.e., undesired, sphericalharmonics generally of the same order and of the degree n,,2, m,,;n,,4,m,,; n,,+2,m,,; n,,+4,m,; with always n2s=m, as well as ancillaryharmonic generations of the same degree and of order 3m,5m, etc., oragain of a different degree of the same parity and of order 3m,5m, etc.The ancillary harmonics generated by one electroconductor interfere withthe primary harmonics generated by other electroconductors and impairthe desired orthogonality. In accordance with a feature of thisinvention, the electroconductors are spatially arranged in a mannerindicated hereinafter, for reducing interfering harmonics at the samplepoint.

The following description exemplifies the detailed generation of a setof shim coils by their projection upon the by their projection upon themagnet pole faces. This set includes 17 primary spherical harmonicpotential functions (mm); (1,0); 1); ,1); ,1); ,1); ,2); (4,1); (4,1);(4,2); (4,2); (4,3); (4,3); and (5,0). This group of 17 functions isconsidered typical for correcting field inhomogenities occurring withNMR magnets. It has beenhoted that sectorial harmonic correction (i.e.,n=m) is not provided since fields which could be corrected by suchharmonics are normal to the field H They have a negligibly smallquadratic effect on the latter when they are smaller than the main fieldby a factor of which is known to be the case for magnets of the typeemployed with NMR apparatus.

The following table I lists these I? harmonics and the correspondingfigures where the locus is plotted and projected. Table 1 lists thesehannonics and their expressions in Cartesian coordinate form. I

The generation of the electroconductors for establishing the sphericalharmonic (1,0) will first be considered. Locus arrangements for theother primary functions and their spatial arrangement will be describedhereinafter. It should be noted that the primary (1,0) function is theonly one of the above functions which does not provide field correctionand thus may be employed for generating a sweep of H as indicatedhereinbefore. However, its generation is typical and the generalprocedure for its generation can be followed for the other four zonalharmonics. In FIG. 5 an electroconductor 72, which is adapted forgenerating this spherical harmonic, is illustrated as being wound aboutan insulative sphere 74. In the FIGS. 5, 7, 9, l1, l3, l5, 17, 19, 21,23 and 25, the sphere 74 is assumed positioned in the magnet gap ofFIGS. l and 2 and the z-axis coincides with the axis of the circularpole faces 10 and 11. The yand z-axes are oriented in the plane of thepaper as shown while the x-axis is normal to the plane of the paper anddirected downwards. The open and solid arrow heads of these FIGURESindicate the direction of current flow in the electroconductor in upperand lower hemispheres respectively of the sphere 74, and solid arrowsare also used to indicate current direction in the xz plane. (In theupper hemisphere, x 0 while in the lower hemisphere x O.) The signs ofthe potential function generated by the currents and shown in FIGS. 5,7, 25 are for the upper hemisphere (.r 0). It is assumed that the centerof the sphere is a unity distance from the pole faces 10 and 11 (i.e.,z=+l and l respectively at the pole faces). FIGS. 6, 8, l0, 12, I4, l6,I8, 20, 22, 24, 26 illustrate the projections of the electroconductorsof the function (1,0) and of other spherical harmonic functions from thecenter of sphere 74 onto the pole faces 10 and II as viewed from the +2direction, i.e., from the right. In several figures inner windings areslightly reduced in size from a strictly linear projection to avoidcrowding. Open directional arrow heads indicate the direction of currentflow in the electroconductor for the projection on face 10 while dottedarrow heads indicate the direction of current flow for the projection onface 11. For the purpose of this specification and the appended claims,the following additional conventions are observed. With respect topolarity, it is stipulated that an observer walking along the winding onthe sphere in the direction of current flow will have negative values ofthe harmonic on his right and positive values of the harmonic on hisleft. Equatorial lines in the x-y plane which would project to infinityon assumedly infinitely extended pole faces are replaced by twosymmetric parallels near the equator which then project at a finitedistance. Only one of each pair (n,m) and (n,m)' has been illustrated,the other one of the pair being obtained by clockwise (c.w.) orcounterclockwise (c.c.w.) rotation of 1r/2m around the z-axis.

It will be observed that the currents flow in the same direction in thetwo windings when n+m is odd, and in opposite directions when n-i-m iseven. This results from the fact that the powers ofz in the potentialfunctions have the polarity of m-l-n, and have the opposite polarity inthe expressions for II which are the derivatives with respect to z ofthe expressions for the potential.

In all cases where m 0, the locus lines which in the figures appear toprovide ajunction point of four leads not located on the equatoractually represent two or a greater even number of separate conductorswhich approach the junction point and turn away immediately beforecontacting. Meridians intersecting the x-y plane at z=0 which wouldproject to infinity, are divided into two segments at a distance fromthe x-y plane at z=0 and continue along the two halves of a parallelcircle, as is illustrated hereinafter for example in connection with theprimary (3,l function electroconductors of FIGS. 13 and 14, so thatthese projections on the pole faces are at a finite distance, asillustrated in perspective in FIGS. 38 and 39. The double arrowsindicate that, in the actual flat coil obtained by projection, two oranother even number of conductors are split and e.g., continue inopposite directions along an arc ofcircle until they meet the projectionof another meridian along which they return. Likewise, when currents onthe sphere arrive from two opposite directions on a meridian, and departin two opposite directions on the equator, as illustrated in FIG. 40,they are split as indicated in FIG. 41, the double arrows having thesame meaning as before.

The projected arcuate electroconductors for a particular harmonic arespaced radially from the z-axis at a pole face in a manner for providingthat ancillary harmonics of the same order but of a lower and/or ahigher degree have a negligible effect at a preselected point in the gapalong the z-axis in the field. That preselected point is the center ofthe sphere which coincides with the sample position.

It will be noted that for reasons of symmetry, the order of theancillary harmonic functions produced by an electroconductor which isarranged for producing a desired primary function is the same as, or, anodd multiple of, the order of the primary functions, and its degree isthe same as, or, differs by an even number from, the degree of theprimary function. Thus, it is possible for ancillary harmonic (3,0) and(5,0) functions, generated by the primary (1,0) electroconductor, toestablish fields causing significant interference with the desiredcorrective fields generated by primary (3,0) and (5,0)electroconductors. In general, the desired spatial arrangement isestablished by determining the magnitude of interference for variousradial spacings of the arcuate electroconductorsand selecting a spacingproviding-tolerably low interference. The magnitudes of these ancillarycomponents and the corresponding spacings are determined from thefollowing analysis. The Biot-Savart law is simplified and writtenvectorially as follows to yield the elementary field dH produced at apoint P by a current unity passing in an element ds of a conductor:

a a? Z where 1 stands for vectorial product, L is the vector from ds toat the origin and neglecting at first the effect of the magneticmaterial of the pole faces:

in which it mustbe understood that z is measured from the pole face,where we have z=0. (This must be contrasted with the fact that we havez=0 at the airgap center for all expressions representing sphericalharmonics. No confusion need arise from this convenient convention.) Theeffect of the magnetic material can be adequately approximated when thepole faces are assumed to be infinitely extended and to have infinitepermeability by placing image loops centered on the zaxis at thedistances z=l ,z=3, z=5, etc., from the gap center. These image loopsare traversed by currents of the same intensit and flowing in the samedirection as flows in the two loops on the two pole faces of the magnet.The successive field gradients in the z direction are similarly obtainedby forming the successive derivates of H with respect to z, from theexpression above, and taking account of the images as indicated exceptfor the factor two due to the image of the loop considered in the poleto which it is immediately adjacent is neglected, as well as anotherfactor two due to the windings on the right pole face. The field writtenabove and its successive derivatives are:

Hill/ 30 These expressions are employed in calculating the zonalharmonics, n,0, generated by the simple loops.

For reasons of symmetry, when the circular electroconductors at the twopole faces are traversed by equal currents having the same direction,they make no contribution to the first and third derivatives, H, andH,", but they do for currents traveling in opposite directions (whilemaking none to H,, H," and H,""), hence the alternation of the signs ofz in the 2's for H and H,

Values proportional to H,, H," and H,"" for different values of r arecalculated in accordance with equations (5), (7) and (9) and aretabulated in table II, and values-proportional to H, and H,'" aretabulated in table III.

TABLE 11 From table II it is seen that both H," and H,"" are relativelysmall when r is'greater than two. Thus relatively small ancillaryharmonics (3,0) and (5,0) will be generated by the primary function 1,0)electroconductor when its radius is greater than two. For purposes ofillustrating a specific example but not deemed limiting in any respect,a magnet with a 2 inch airgap, an electroconductor width of the order of0.0075 inches, and an insulating space between electroconductors of theorder of 0.005 inches is selected. With due regard to minimizing currentamplitude requirements, then four generally circular arcuate segmentsha'ving radii selected to be 2.10 inches; 2.l4 inches; 2.18 inches; and2.22 inches are formed. This electroconductor; having a differing scalefor purposes of clarity in the drawings, is illustrated in FIG. 3 andthe arcuate segments are indicated by the reference numerals 76, 78, and82 respectively. But henceforward the convention indicated earlier ofmaking the distance from the airgap center to either pole face will beadhered to, and the several radii selected for several arcuateconductors will be shown as numbers.

The locus of electroconductors fonned on an insulative sphere andadapted for generating (2,0); (3,0); (4,0); and (5,0) primary functionsare illustrated in FIGS. 7, 11, I7 and 25 respectively.

Projection of the locus of these functions also results in circularelectroconductor arrangements. Interference with the primary L0) and(5,0) functions by the ancillary functions of the (3,0) windings, is tobe avoided as well as interference with the primary (1,0) and (3,0)functions by the ancillary function of the 5,0) winding.

Anticipating the use of other electroconductors for other harmonicorders as indicated hereinafter, and in view of H, as calculated, thefollowing positions are selected for the primary (3,0) functionelectroconductors:

a. for the electroconductors projected on pole face 10 and conductingclockwise flowing currents in FiG. l2, r=0.78, 0.82, 0.94 and 0.98; andhaving the average position r=0.88; and r ing the zonal functions (1,0),

b. for the electroconductors projected on pole face 10 and conductingcounterclockwise flowing currents in FIG. 12, r=1.66, 1.70 and 1.74; andhaving the average position r=l .70.

The interpolated values of H, at r-0.88 and 1.70 from table II are 0.38and 0.49 respectively, values which when multiplied by four and threerespectively, in order to reflect the respective number of loops nearr=0.88 and r=l .70, and subtracted, give an adequately small residualfor contamination of the primary 1 ,0) function by the ancillary 1,0)harmonic of the primary (3,0) function. Likewise, the interpolatedvalues of I-l,""at r=0.88 and 1.70 are --0.01 and 0.03, and those smalland partially cancelling values indicate that the primary (5,0) functionis also substantially free of (5 ,0) contamination from the primary(3,0) function.

The position of the primary (5,0) function loops were similarlydetermined as follows:

a. for the inner electroconductor projected on pole face and carryingclockwise flowing current in FIG. 26, r=0.34 and 0.38; i.e., two loopsat the average position !=0.36;

b. for the intermediate electroconductor projected on pole face 10 andcarrying counterclockwise flowing current in FIG. 26, r=0.86 and 0.90;i.e., two loops at the average position r=0.88, and

for the outer electroconductor projected on pole face 10 and carryingclockwise flowing current in FIG. 26, a single loop at r=l .78.

The interpolated H, contributions at the three average r values juststated are 0.1 l; 0.36 and 0.49 which, when multiplied by (2); by (2),and by (l) and added indicate a near arithmetic cancellation, so thatthe primary (5,0) function electroconductor placed as indicated above issubstantially free of ancillary (1,0) which would contaminate theprimary (1,0) function. The interpolated values of H," at r=0.36; 0.88and 1.78 are 0.33, 0.34 and 0.64 which, when multiplied by (2), by (2)and by l) and added give the small residual 0.02 which indicates thatthe primary (5,0) function is also substantially free of ancillary (3,0)which would contaminate the primary (3,0) function.

As illustrated in FIGS. 8 and 18, the primary (2,0) function and primary(4,0) function electroconductors conduct current in opposite directionson the two pole faces 10 and 11 and their radial spacing is calculatedfrom the alternating series (6) and (8), for which the following valuesare computed:

TABLE III For the primary (2,0) function and with due regard to reducingthe current amplitude required, three loops are selected at r=l.l0; 1.14and 1.18, i.e., at an averagevalue of r=l.l4 for which value of r, H,'is negligibly small as interpolation indicates.

Similarly, for the primary (4,0) function, two clockwise loops areselected at r=0.42 and 1.46, and two counterclockwise loops at r=l .38and 1.42. It is seen from the calculated values of table III that H,interpolated at the average positions =0.44 and 1.40 is 0.12 and 0.115respectively, indicating here, also, the near freedom of the (4,0)windings from (2,0) contamination. The electroconductors for generat-(2,0), (3,0) and (5,0) are illustrated in FIG. 3, and theelectroconductor for generating the zonal function (4,0) is located onthe second of the two pairs of plates mentioned earlier, and isillustrated in FIG. 4.

The electroconductors for the order m=l are formed, as described indetail hereinafter, by the superposition of semicircles. A semicircle200 as shown in FIG. 27 is positioned at both pole faces 10 and 11, andis paired with a second semicircle symmetric with respect to the z-axisand traversed by current in a direction opposite to the direction ofcurrent in the first semicircle in a manner for providing that thecontribution of the two paired semicircles to the field H, will cancel,while their contributions to the field H will add.

A primary (1,1) function electroconductor, although not one of the 17primary functions being generated, would produce the potential:

=y (15) the derivative of which with respect to y is unity, and thederivatives, Hy, Hy" and Hy' of the potential actually produced by a(1,1) winding will be measures of the (2,1 (3,1 and (4,1 functions whichit may be desired to produce, or not to produce, by several of theelectroconductors intended to produce harmonics of first order (m=l Anexpression for H, can be derived by applying the Biot- Savart law toobtain the contributions to H, of the radial segment 202 and curvedsegment 204 of the semicircular electroconductor illustrated in FIG. 27.

The contribution H of the radial segment 202 is proportional to theintegral:

The contribution H of the curved segment 204 is proportional to theintegral:

The H,, field produced by a semicircular electroconductor 200 and by theother obtained by imaging with respect to the .rz plane, is then givenby the expression:

When the images of these electroconductors in pole pieces of assumedlyinfinite permeability are taken into account, the following are usefulquantities for determining the electroconductors for producing harmonicfunctions of the first order in the y or x direction, where thesuperscript prime symbol again designates differentiation with respectto z:

one-fourth of H and one-twelfth of H' (also designated hereinafter as A)are calculated for various values of r as indicated in table IV:

Table IV will be utilized to establish primary (2,1 functionelectroconductors with reduced interference from their ancillary (4,1)function with the primary (4,1) function and for establishing primary(4,1) function electroconductors with reduced interference from theirancillary (2,1) function with the primary (2,1 function.

It is noted that all but the zonal harmonic electroconductors requirewindings with segments extending through the z-axis. Since theaforementioned plate configuration will accommodate conveniently onlyfour straight segments, i.e., two side by side, on each of the two sidesof the plate, circumferential detours are provided in order to avoid theextension of a straight segment through the center. These detours willbe arranged to avoid certain contaminations. A typical detour will bemade for the primary (2,1) function electroconductor, one-half of whichis illustrated by FIG. 28, the other half being symmetric with respectto the horizontal line. FIG. 29 illustrates that the actual (2,1)electroconductor selected can be considered as the combination of two ofthe single windings for which H,," has already been calculated above.This arrangement will be utilized generally to avoid severalcontaminations.

It will be noted also that the physical necessity of providing severalcurrent paths which are parallel to each other in the radial directionshas demanded that the semicircular or sectorial building blocksillustrated in FlGS. 29, 30, and 32 out of which all windings are formedbe made somewhat smaller, with their straight portion displaced towardstheir centers, while the circular portions are still centered on thez-axis. It has been verified by actual calculations that in all theexamples shown here the effect of these departures from the idealconfiguration is to decrease the fields produced by a few percents only,and since the ratios of these fields, which are of primary concern, areaffected by the differences between these few percents, this effect hasnot been taken into account in the calculations shown below, but couldindeed be reckoned with if extreme mathematical precision were requiredin the designs. I

In the case of the primary (2,1) function electroconductor, shown inFIG. 4, the interpolated values of H,,"' at r=0.54 and r=0.58 are 2.20,and 2.37 respectively, and the interpolated values of H,," at r=1.34 andr=1.70 are 2.42 and 2.15 respectively, and since there are foursemicircular electroconductors at the first two radii given above, andfour semicircular electroconductors with the opposite direction at thelast two radii given above, it is seen that, because we have:2.20+2.37=2.42 +2.15, we have an excellent cancellation of the unwantedancillary (4,1) function, hence the selection of the radii given abovefor the arcuate segments of the (2,1 electroconductor. Likewise, theradii of the 2,1) function selected were 0.50 and 0.62 for onedirection, for which radii H,,"' has the respective values 2.00 and 2.51respectively, and 1.46 and 1.58 for the other direction, at which latterradii H,,"' has the respective values 2.31 and 2.20. Since we have:2.00-+2.51= 2.31+2.20, it is seen that the cancellation of the unwanted(4,1 harmonic for the (2,1) electroconductor is also excellent.

The freedom of (2,1 contamination of the field generated by the primary(4,1) function electroconductor is likewise effected. FIG. 30 indicatesthat the primary (4,1) function can be composed of several semicircles.At the r=0.68, and 1.34 positions, the interpolated values of H, for theancillary (2,1) function are 0.55 and 0.09 which, when multiplied by (2)and by (-l) respectively, and added substantially cancel. Accordingly,the primary (4,1 function electroconductor is assigned the midconductorradii r=0.62, 0.74 and 1.34. The two center positions for the doubleelectroconductor shown running counterclockwise on FIG. 30 have theaverage value 0.68 indicated above, and are spaced apart to make roomfor the primary (4,1) function electroconductor. The latterelectroconductor was assigned the radii r=0.66, 0.70, which have theaverage value I==0.68, and 1.30, for which the interpolated values of H,are: 0.55 and 1.08 which, when multiplied by (2) and by (-1 and added,substantially cancel.

Sincethe ancillary (1,1) and (3,3) harmonic functions also generated bythe (3,1) winding are perpendicular to the main field, they are ofnegligible importance. The three (3,1

semicircular segments were assigned the radii 0.90, 1.02, and 1.82.Similarly, the (3,1) function electroconductor was assigned themidconductor positions r=0.94, 0.98 and 1.78.

For the (4,2) electroconductor the following radii were selected:r,=0.34, r =1.l2 and r =2.22. As it is, it can be calculated that littlecontamination of the (2,2) harmonic is present, but this is of littleimportance as the corresponding magnetic vector is perpendicular to themain field. Similarly, the following radii were selected for the (4,2)winding: r,= 0.38, r 1.12 and r =2.l8 for which there is also little(2,2) contamination.

No avoidance of contamination is sought for the primary (3,2), (3,2),(4,3) and (4,3) function electroconductors. The contamination producedby these windings are of a higher degree and are considered negligible.The following midconductor loop positions were assigned to theprojection of these electroconductors:

(3,2) r=0.22 and 1.22 (3,2) I=0.26 and 1.26 (4,3) r=0.18, 0.30, 1.02 and1.26 (4,3) r=0.22, 0.26, 1.06 and 1.22

It is necessary, however, to avoid contamination of the 4,3) winding bythe (2,1) and (4,1) windings on the one hand and of the (4,3) winding bythe (2,1) and (4,1) on the other hand. This is accomplished inaccordance with a feature of this invention by a network meansillustrated in FIG. 34. In FIG. 34 the potentiometers 300, 302 and 304are of negligibly small impedance. An arm 299 of the potentiometer 300is adjusted to control the current amplitude in the (2,1 winding. Thiscurrent is not affected by the adjustment of the potentiometers 302 and304. Similarly, an arm 301 of potentiometer 302 controls the current inthe 4,1 winding without contamination by the other currents. An arm 303of potentiometer 304 controls the current in the (4,3) winding. However,in the latter case the current in the (4,3) winding is also affected,through resistances 314 and 316, by the settings of the arms ofpotentiometers 300 and 302 respectively. The order of the windingcorrections are calculated in a manner for providing that the physicalcontamination of the (4,3) field by the (2,1 or 4,1) windings issubstantially correct. The determination of this combination proceeds asfollows. The simple semicircular loop of FIG. 27, traversed by currentunit, together with the semicircle symmetric withit with respect to thexz plane and the other two semicircles symmetric with these with respectto the xy plane produce some (4,1 and (4,3) spherical harmonics whichcan be written with generality as:

=.1yz[4z 3(x-l-y }+Byz( 3x y (22) The Biot-Savart law, simplified asindicated earlier, is utilized for the determination of the H, vectorproduced by the semicircle of FIG. 27 and the result is multiplied bytwo to account for the contribution of the other semicircular loop. Forunit current, and inclusion of the images of the current loops in thepole faces, we have:

and, as tabulated hereinbefore in Table IV from which i6 complexnetworks required for field homogenization when the NMR sample is spun.

The foregoing described the generation of orthogonal, current-traversedwindings for the purpose of cancelling the inhomogenities ofa magneticfield in which an NMR fixed sample is studied.

When the sample is spun, e.g., around the y-axis, the H field for anyone nucleus is averaged along a circle centered on the y-axis and in aplane normal to it and the central absorption line of a generatedspectrogram will be sharpened correspondingly. The only remaininginhomogenities acting on this line are those which are due to thevariation of the field average with y, and which can be expressed asspherical har- TABLE V monies written as functions ofy and ofx +z suchas y, Zy Lr 1 +2 2y -3.r(x l-z etc. On the other hand there will be in rIA 1 V2 w addition to the main central line, frequency modulation side-23 bands spaced from it by the rotational frequencyfof the sam- W 7 Hple and multiples thereof. A spectrogram of this type is illustrated inFIG. 35. For example, there will exist at the frequen- Likewise th li if h Bi s law f the wind cy interval fat f sides of the central line,sidebands due to ining of FIG. 33, where the double arrows denote doublecurhomogenities Proportional to X or 3 Such as those for which rent,gives for that winding and the other one obtained by the field belwrmenagain imaging in the xz plane, and including the images in the pole ifbe at dlsumfe f 2f on either side of the central lrne, sidebands due toInhomogenities of H which can be written (x -z), Y(x z etc., or .rz,xyz, etc., and so forth for higher frequencies.

Most of the magnetic potentials which, when differentiated o 168 4 5+812 7+1 9 with respect to z, produce the expressions written above are m z3 5 z4(z2+7.z) 9/2 not the spherical harmonics listed in table I but arelinear combinations of those spherical harmonics. If the windings (26)described hereinbefore for homogenizing the magnetic field T for astationary sample are retained, with their individual current-adjustingcontrols, the operation of one such control afwhere the asterisk isutilized to denote the circumstance fects simultaneously the amplitudeof the central line and sidethat the calculations are carried out withrespect to the bands of FIG. 35, or several sidebands, while the centralline windings of FIG. 33 (as against the simple semicircularwindamplitude is affected by several windings acting redundantly ingused earlier), and where it will be noted that the value obh respect oea h o h r. tained is three times the contribution to 68 obtained i (23)In order to restore orthogonality of settings, it is necessary for thesimple semicircle of FIG. 27 and its images, and written to express thha monic functions o d gre n higher than for 2B in (25). Thiscorresponds neatly to the fact that, taking from which the expressionswritten above are obtained by difh semicircle f I 27 and its images plusthis ensemble ferentiation with respect to z, as linear combinations ofthe rotated at 120 around h taxis, reproduces exactly the sphericalharmonic functions referred to the polar z-axis. For figuration of FIG.33 a d it i this purpose, table VI has been formed for each degree ofthe From thi it i ibl t interpolate f bl V d spherical harmonicsstarting at the second and including it for culate the contamination ofthe (4,3) a d th (4,3) i di the sake of completeness. The first column,entitled frequenby the ancillary functions produced by the (2,1) and(4,1) ricy, indicates whether that harmonic of interest which maryfunctions, on the one hand, and by the (2,1) and (4,3) represents aninhomogenity affects the central line (C) of the on the other hand; todivide by three and compare with the spectrogram or a sideband spaced f,2}", etc., from a central (4,3) and (4,3) interpolated B values; and todetermine suitaline. All the js are paired corresponding to quantitiesproporble magnitudes of the resistances 310 through 320 of FIG. 34tional to the cosines (x, (x z etc.) or the sines (z, xz, etc.) as wellas the proper current directions so as to correct for of once, twice,etc., the angle measured from the x-axis, as a contaminations. The kindof calculations involved here will be line starting on the x-axisrotates in the x-z plane around the yillustrated below in greater detailin connection with the more axis.

Freq Hz Magu. pot. Decomposition Symbols Second degree 11 (2, 1) :2 I:22- (x +y 22 (z +1/ Third degree +r/ )l+% v %(3,0)+% (3, l/ 21/: (3, 21/[ +r/ )l ra -WHO] 1 z +Y -U %(3, :c[4z (z +y)] :[42 (Mimi 1) Fourthdegree 0 2y=3y(z +z yz +3yzz 2u=z 54y:[42 -3(z +y )]+91/.z(3z u) i;)'+ fm -WHO] z2 +3I 212zz/ z %zz[4z 3(z +u 4zz(z 3y 4',1)+ I 2[4u (z +z z +2zz 8z u %4z +%z u 94u y[8z 24z (:c )+3(:r: +1 +%(z y )[6z (z +1/ )l $66:534-9 2f MI tie-3x 11: ;4 z 4z -a z +w ow- %(;:1; 2f u z 2 3 3 z 2 2 1ar -an S e -"m y fiifiw aiii zw-an) 9-34 m4, o -;4 4, 2

The second column, entitled FL, contains the expressions for H whichcorresponds to CJ" and f, etc., for the ascending degrees. The thirdcolumn indicates the form of the magnetic potentials from which H, canbe derived and in which the terms not containing 2 have been so selectedthat these potentials can be expressed in terms of the zonal andtesseral harmonies with the polar z-axis only, i.e., without sectorialharmonies, as this can be gathered from the forth column. The fifthcolumn contains the shorthand expressions for the fourth column inaccordance with table I.

It will be noted from column five of table Vl that several sphericalharmonics occur singly and only once, and, therefore, need not behybridized. It will be further noted that several occur in pairs, and ineach instance twice. Two pairs, namely (3,0) and (3,2), hybridized.Since each pair occurs once as a sum and once as a difference of twoharmonics, the hybridization can be effected by means of a circuit suchas the one shown in FIG. 36, while the relatively more complicatedcircuit of FIG. 37 provides for the hybridization of (4,1) and (4,3), onthe one hand, and the hybridization of (4,l and (4,3) on the other,because of the respective contamination of the (2,1 and (2,l winding.

Referring now more particularly to the 3,0) and (3,2) harmonics, columnfive of table Vl indicates that their relative values should be in a 1:9ratio in one case, and in a l :3 ratio in another. However, in order toobtain the current ratios, ad ditional weighting factors must beintroduced in accordance with the values of H," of table II for the(3,0) harmonic, and in accordance with the values of which are obtained,in turn, from the values of b Hy 02: oz

Several values of the bracketed term (27) are tabulated in table Vll.

TABLE VII 1 9 4 1W 1 1/ 2 m dzHv m +.12 +.36 +.82 1-43 1.89 +2.06 +2-11Let now the harmonic required for the correction of the third degreeharmonic which affects the central line or the Zfsideband be written:

=ai(3,0) (3,0)+i(3,2)-(3,2) (28) where i( 3,0) and i(3,2) designate thecurrents circulating in the 3,0) and (3,2) loops respectively and a and,8 coefficients to be determined.

We have, for current unity in the (3,0) loops T=1202H, where the sumextends over all the loops allotted to the 3,0) winding. The H, s areobtained from table ll and the equation above determines a as indicatedin the calculations which follow in which the direction of the currentin the various loops was selected so as to make a and ,6 positive. Thesame device has been the other networks.

and (4,0) and (4,2) can be simply is (3, 0) r= values 2H,: values Wehave also for the loop of FIG. 31 and its three images, when rotated 45clockwise around the z-axis when viewed from +2:

and since the loop so rotated and traversed with current unity producesthe potential Bz(x"'-y), before rotation it produces Zlixyz, from whichwe obtain:

0 (25:01:12) b Hg bxbybz bx a2 (31) where the summation extends over allthe loops allotted to the 3,2) winding, the

being obtained from table determines B as indicated below:

(3, 2) r= values values c.c.w. loop L26 1.85 c.w. loop 0.26 0.()5 TOTAL1110 whence: 2B=4( 1.80) and B=3.60 From table Vl, last column and C and2f lines of the third degree tabulation, we obtain:

az'(3.0) {1/9 for (0) control flz'(3.2) 1/3 for (2f) control or, sincea/B=+l .93/3.60 =+O.536

i(3.0) {+.207 for (0) control z(3.2) .62 for (2f) control Th e T-networko t l iiia adeEimeTyTof these conutilized in all similarcalculations below for trols, provided the C, coil is made the (3.2)coil and the C coil is made the (3.0) coil so that, when V is, e.g.,positive, the current flows in the loops of (3.0) and of 3.2 asindicated in the tabulated calculations above and provided also the V,control be assigned to (C) and the V, control to (2]). Then, when the(C) control is operated, incremental +Ai(3.0) and Ai(3.2) values shouldbe in the ratio:

e (51385501 is operated,

tiometers of adequately small resistance, and convenient

1. In an NMR apparatus having means for establishing a magnetic field ofpredetermined magnitude in a gap between first and second magnet polefaces, a means for reducing inhomogeneities occurring in the magneticfield comprising: first and second plates formed of an electricallyinsulating material positioned in the gap near said first and secondpole pieces respectively, said plates each supporting a plurality ofspaced-apart electroconductors having segments thereof formed onopposite surfaces of plates, each of said electroconductors adapted toestablish a magnetic field representable by a distinct sphericalharmonic function when a current flows therein and circuit means forcausing a current to flow in each of said electroconductors forhomogenizing said field.
 2. The apparatus of claim 1 wherein saidelectroconductors comprise generally arcuate and radial segments.
 3. Theapparatus of claim 1 wherein said electroconductors are formed ofrelatively flat generally arcuate and radial segments and are positionedin a manner for providing orthogonality between primary sphericalharmonic functions (np, mp) which are representative of the magneticpotentials generated by said electroconductors.
 4. The apparatus ofclaim 3 wherein said homogenizing electroconductors comprise circularsegments for providing a primary spherical harmonic potential functionof degree n and order m
 0. 5. The apparatus of claim 3 wherein saidhomogenizing electroconductors comprise arcuate segments arranged as aplurality of generally semicircular segments connected to radialsegments for generating a primary spherical harmonic potential functionof degree n and order m
 1. 6. The apparatus of claim 3 wherein saidhomogenizing electroconductors comprise arcuate segments arranged in aconfiguration of a plurality of generally quarter-circular segmentsconnected to radial segments for generating a primary spherical harmonicpotential function of degree n and order m
 2. 7. In an atomic analysisapparatus having means for establishing a magnetic field ofpredetermined magnitude in a gap between first and second pole faces, ameans for reducing inhomogenities in the magnetic field comprising:first and second plates positioned in the gap near said first and secondpole faces respectively; said plates each comprising an insulativematerial having a relatively thin conductive material mounted onopposite sides of the insulative material; said thin conductivematerials on each of said plates forming a plurality of electricallyinsulated electroconductors adapted to establish an equal plurality ofcorrective magnetic fields which are defined by primary sphericalharmonic potential functions of order m 0, 1, and 2; saidelectroconductors including arcuate segments and spaced relatively in amanner for providing substantial orthogonality between fields thepotentials of which are primary spherical potential functions (np, mp);said arcuate segments for establishing a primary spheRical harmonicpotential function of order m 0 arranged in a generally circularconfiguration; said arcuate segments for establishing a primaryspherical harmonic potential function of order m 1 arranged in aplurality of generally semicircular segments connected to radialsegments; said arcuate segments for establishing a primary sphericalharmonic potential function of order m 2 arranged in a plurality ofgenerally quarter-circular segments connected to radial segments; andmeans for causing current to flow in each of said electroconductors forhomogenizing said field.
 8. The apparatus of claim 7 wherein the arcuatesegments are interconnected by radial extending segments formed from theconductive material on the opposite surface of the insulative material.9. The apparatus of claim 7 including additional pairs of platesarranged for providing additional primary spherical harmonic potentialfunctions.
 10. In an atomic analysis apparatus having magnet means forestablishing a magnetic field H1 of predetermined magnitude in a gapbetween first and second pole faces of the magnet, a means for reducinginhomogenities in the magnetic field comprising: first and second platespositioned in the gap near said first and second pole facesrespectively, said plates each supporting a plurality of electricallyinsulated electroconductors formed in a same plane, each of saidelectroconductors having a substantially uniform cross section andadapted for establishing a distinct corrective magnetic field thepotential of which is representable by a spherical harmonic function; aone of said electroconductors adapted to establish a magnetic fieldpotential representable by a primary spherical harmonic function ofdegree n 1 and order m 0; and, circuit means for causing currents ofseparately adjustable magnitude to flow in each of saidelectroconductors for homogenizing said field H1.
 11. In a nuclearmagnetic resonance apparatus having means for establishing a magneticfield of predetermined magnitude in a gap between first and secondmagnet pole faces, a means for reducing inhomogenities occurring in themagnetic field comprising: a plurality of windings positioned betweensaid pole faces, each of said windings arranged for establishing aprimary magnetic field representable by a distinct spherical harmonicfunction of particular degree and order and simultaneously creatingancillary magnetic fields in said gap when a current of predeterminedamplitude flows therein; first circuit means for providing a separateelectrical current flow path between each of said windings and a currentsource for homogenizing said field; a current flowing in a firsthomogenizing coil of a first electrical current flow path creating anancillary magnetic field which interferes with a primary magnetic fieldcreated by a current flowing in a homogenizing winding of a secondelectrical current flow path; and an electrical impedance coupledbetween said first and second current flow paths for applying a portionof a current flowing in said first path to said second path, saidimpedance having a magnitude for applying a current to said secondcircuit having an amplitude for cancelling the interference of saidancillary field on the primary field of said second circuit winding. 12.The apparatus of claim 11 wherein said plurality of homogenizingwindings includes windings adapted for generating homogenizing fieldsthe potentials or which are combinations of the spherical harmonicfunctions of the degree and order (2,1), (4,1) and (4,3), and saidelectrical impedance means is arranged for linearly adding currentsproportional to currents flowing in the (2,1) and (4,1) windings to thecurrent flowing in the (4,3) winding to reduce interference of fieldsgenerated by the former on the field of the latter winding.
 13. TheappaRatus of claim 11 wherein said plurality of windings includeswindings adapted for generating homogenizing fields the potentials ofwhich are combinations of the spherical harmonic functions of the degreeand order (2,1)'', (4,1)'' and (4,3)'', and said electrical impedancemeans is arranged for linearly adding currents proportional to thoseflowing in the (2,1)'' and (4,1)'' winding to the current flowing in the(4,3)'' winding to reduce interference of fields generated by the formeron the field of the latter winding.